Tente algo assim:
#!/bin/bash
if pgrep "mysql" > /dev/null
then
echo "MYSQL Running"
rm -f /var/run/.mysql_mail_sent
else
echo "ALERT Stopped"
if [ ! -f /var/run/.mysql_mail_sent ]; then
// send your mail here
date > /var/run/.mysql_mail_sent
fi
fi